368
THE THEORY OF THE HELMHOLTZ RESONATOR
With this alteration (15) holds good for the interior, U denoting localised normal velocity at the surface still measured outwards, U= dty/dr.
We have now to introduce approximate values of %7l_i (c) -4- %n (c) in having regard to the assumed smallness of c, or rather kc. For this pu: we expand the sine and cosine of c* :—
cos c __ 1___c_ >&__&_,
1 d_ /cos c\ _ !_ 1 3c 5c3 __ 7c" "efcl~c~/~c8 1.2c 4! 6! 8! + >>1'
r cos c _ 3 1 3
c dc) ~~cT = o5 + 1. 2c3 c751
5.3.C 7 ._5_. c;i "6! + " 8'""
and so on;
sne __ c?_ c* _ c^ ~~T" ~273 5! 7!
1 d sin c 2 4c2 6c
1
i
snc
\ c dc) c 5!
and so on. Thus for the outside
1.3.5...(2??.-1)
7!
For general values of n, we may take
For n=
1 c .
c 2~*
+
9!
1-
2(2n + 3) r '"} '
= -^+ terms in c4. "
(1
.(i
(1!
* 1917. In the expansions for the derivative of cos c/c terms (now inserted) were acoider omitted, as has been pointed out by Mr F. P. White (Proc. Roy. Soc. Vol. xcn. p. 549). Equ (17) as originally given was accordingly erroneous. Corresponding corrections have been i duced in (19), (23), (24), (36), (38) which however do not affect the approximation employed in Mr White's main object was to carry the approximation further than is attained in (57) and (es the value of ^ at a point whose angular distance (0) from da is cos"1/A. f XH has the form given by (4), the result applies to the exterior surface f the sphere.